Probability Definition and 1000 Threads

I want to find the probability that the two points ($x_1, y_1$) and ($x_2, y_2$) lie on the opposite sides of a line passing through the origin $o = (0, 0)$ and makes an angle $\psi$ that is uniformly distributed in $ [0, \pi]$ with the $x$ axis when the angle is measured in clockwise direction...

I have been looking for a way to learn probability and statistics online and have searched but found nothing yet. I am looking for a course on probability and statistics that will not only teach me the basics but all there is to know about the subject. I would love it if the visualizations are...

Hello, everyone. The projection postulate says the wave-function collapses to one of its eigenstates under measurement, does it talk about each probability with which the wave-function collapses to those possible eigenstates?

Starting with finding the probability of getting one of the states will make finding the other trivial, as the sum of their probabilities would be 1. Some confusion came because I never represented the states ##|\pm \textbf{z}\rangle## as a superposition of other states, but I guess you would...

"Prove Theorem 7.1 about the probability of a union, using the 12.3 proof (see section 12.2) that involves indicator variables. Do not write the proof in full generality, only for three events. You should not use the product notation; you should write out all factors of the product." I'm taking...

Hi PF, A(x,t) is the probability amplitude at time t that a particle is at x. If it was emitted at (0,0) the propagator gives its value. I wonder if QM can give the amplitude of time probability B(y,t) that an impact will occur (for a given y) at any t. consider a screen behind the two slits, it...

1) Since I want at least ##6## flights to come within ##2## hours, then the time interval between each should be, at worse, ##2/6=1/3## hours, and the probability is ##P(X\leq1/3)=1-e^{-1/3}##. 2) The probability that at best 5 airplanes arrive at the airport is...

Suppose P(B|A)=1. Does that mean that P(A|B)=1?

Summary:: How to solve this? See attached. This is from Griffiths 3rd edition quantum mechanics textbook, problem 2.7: This is the solution for an infinite square well. I am to find the probability of obtaining E1 where En = n^2pi^2hbar^2/2ma^2. E1 = hbar^2pi^2/2ma^2 When finding the...

Can someone help me on this question? I'm finding a very strange probability distribution. Question: Suppose that x_1 and x_2 are independent with x_1 ~ geometric(p) and x_2 ~ geometric (1-p). That's x_1 has geometric distribution with parameter p and x_2 has geometric distribution with...

If my partition function is for a continuous distribution of energy, can I simply say that the probability of my ensemble being in a state with energy ##cU## is ##e^{-\beta cU} /Z##? I believe that isn't right as my energy distribution is continuous, and I need to be integrating over small...

Given ##X_1,X_2,X_3## three independent random variables that can assume values 1 ,0 ,-1 with probabilities ##\frac 1 4 ,\frac 1 2 ,\frac 1 4 ##. Let ##S=X_1,X_2,X_3## and let N be the number of##X_i## assuming value ##0##. Knowing that N=1 i have to find what os the probability of ##X_1=0##...

So first i compute the number of manners i can choose a sett of 3 ppl among 6 ##\frac {something}{\binom {6}{3}}## but then i don't understand from where should i pick up the two friends, from the set of those going to the party or what? Sorry but have problems on understanding how to deal...

My reasoning is : first i evaluate in how many manners i can choose 2 guys from the group of 8. ##\binom{8}{2} \binom{6}{2} \binom{4}{2}\binom{2}{2}## then i consider the probability of choosing the seventh floor is 1/4. But now I don't now how to proceed, supposing up to here i am correct...

Question 1: a) T' is the complementary event of T Therefore, T'=1-T In set T = {3,6,9,12} P(T)=4/12 =1/3 P(T')=1-1/3=2/3 b) The addition rule states; P(A ∪ B)=P(A)+P(B)-P(A⋂B) Therefore, P(S ∪ E) = P(S)+P(E)-P(S⋂E) S={1,4,9} P(S)=3/12=1/4 E={2,4,6,8,10,12} P(E)=6/12=1/2 (S⋂E)={4} P(S⋂E)=1/12...

A single electron sitting in a void has an electric field that spreads out evenly in all directions as far as there is open empty space to allow it, is this roughly a correct statement? Let's say we now introduce a singe proton into the void, 100 miles from the electron - it will also have an...

Let's say I'd like to calculate the probability of getting at least one 4 when rolling two dice. That's 1 minus the probability of not getting any 4, i.e 1 minus the complement, 1-(5/6)^2. But how would I calculate without using the complement?

Say I make it so that the 2 coin flips count as a single number 1,2,3,4 representing head-head, head-tails, tails-head, tails-tails. Then what do I do? I'm just lost as to how I would even approach this problem.

Confused and not sure if it is correct, but please do correct my steps. We let event B be that there are at least 3 customers entering in 5 minutes. Hence P(B) = 1- P(X=0)- P(X=1) - P(X=2) = ##1- \dfrac{e^{-5}5^{0}}{0!}-\dfrac{e^{-5}5^{1}}{1!}-\dfrac{e^{-5}5^{2}}{2!} ## = 0.8753... Now we let...

I thought this was kind of a cool proof of the product rule. Let ##F(x)## and ##G(x)## be cumulative distribution functions for independent random variables ##A## and ##B## respectively with probability density functions ##f(x)=F'(x)##, ##g(x)=G'(x)##. Consider the random variable...

Summary:: When filling up carts with iron the real weight deviates from the nominal value 10 ton. The standard deviation is .5. What's the probability that 25 carts exceed 255 ton? The arithmetic median value is: $$X \in N(25 * 10, 0.5 / \sqrt{25}) = N(250, .1)$$ $$P(x > 255) = 1 - P(x < 255)...

For the region where V = 0, solving the schrodinger equation leads to the above value of wave function, psi = sqrt(2/L) sin(pi x/L) Since in the qus. it is not stated about the 'direction of movement' only restricted to +x direction, I think that the probability will be 1/2. And finding the...

Hello, In probability math,because of math's nature that is merely quantitative and not a qualitative, for any case,it give you just a number; so, I think for every cases, there should be a boundary probability number that is " meaningfulness " just for that specified case and out of that...

Hello! I'll be taking a probability and statistics course this semester. Does anyone know of any good textbook? I have access to an extensive catalogue of books on springer, so it would be extremely preferable for me if you could recommend something from there. Thanks.

In Scientific American, July 2020, the article "The Darkest Particle" by Louis and Van de Water, page 46, discussing the hypothetical sterile neutrino, there is the sentence: "Because sterile neutrinos are likely to be more massive than the regular flavors, however, particles could make the...

I don't see why it is not ##P(\omega)\propto |\langle \psi | \mathbb{P}_{\omega}|\psi\rangle |^2.## After all, the wavefunction ends up collapsing from ##|\psi\rangle## to ##\mathbb{P}_{\omega}|\psi\rangle.##

Can MWI account for the probabilities of outcomes? If MWI says all outcomes are realized, is the probability that an outcome occurs then not 100%? How is this explained with the entanglement of the measured object and the measurement apparatus?

I have the following probability density function: $$f(x) =...

Here is my work so far: 52-13=39 There are 39 decks of cards left since the spades were removed. a) Then there 13 hearts therefore, (13/39 ) * (13/39 ) that would be two hearts but I am not sure what to do next. But I am sure that it would be 39 cards and 13 hearts on top (maybe) cus it is...

This questions was brought to my attention by Kazu Okayasu. According to probability theory the probabilities of mutually exclusive events add upp. As an example we can distribute 2 balls in two boxes with two compartments each. So there's a box on the left with a lower and an upper compartment...

I have the following probability density function (in Maple notation): f (x) = (1 / ((3/2) * Pi)) * (sin (x)) ** 2 with support [0; 3 * Pi] Now I want to transform x so that 0 -> (3/2) * Pi and 3 * Pi -> (15/2) * Pi and the new function is still a probability density function. How should I...

In the keq derivation intuition video mr sal Khan relates molarity to probability and it doesn't makes sense to me as molarity can also be more than 1 and probability cannot. Can you please tell me how he relates probability to molarity. Thanks

Given the support [a, b] of a probability density function. How can I change the formula for the probability density function with a support [u, v]? Example: Given the beta distribution with support [a=0,b=1]: $$\frac{x^{p-1} (1-x)^{q-1}}{Beta(p,q)}$$ Then the beta distribution with support...

Please help me solve this, Given digits 2,2,3,3,3,4,4,4,4, how many distinct 4 digit numbers greater than 3000 can be formed? Thank you

Two-marble probabilities BB 1/4 BW 1/2 WW 1/4 Three-marble probabilities of selecting a black B BB 1 B BW 2/3 B WW 1/3 Probability of selecting a black marble if we add a black marble to the two marble bag (1 x 1/4) + (2/3 x 1/2) + (1/3 x 1/4) = 2/3 Three marbles with 2/3 probability of...

From the probability density plot why it is max at the centre of the box...e.g probability to find the particle per unit le gth is max. at the centre of the box??why not at any other position??

We're playing a game with regular, fair, six-sided dice, and we're throwing 14 of them. The "1" is a "joker", which means it can count as any other number. So, for example, if we roll three "1"s and four "2"s, we have seven "2"s. What is that probability that we get exactly seven of any face...

The chance of everyone finishing should be $$Fx(100)^8 = (100 * 125/450 - 100^2/900)^8 = (50/3)^8$$ What am I doing wrong?

A dice was thrown for 6 times.What is the probability that at least 5 was shown for 5 or more times?

My questions are as follows: 1. How do we find them and why do we need them? 2. What are the meanings of the mean and the median of a PDF? Are the formulae below correct? $$\int_{a}^{median} f(x) \mathrm{d}x = \int_{median}^{b} f(x) \mathrm{d}x$$ $$\int_{a}^{mean} f(x) \cdot x \mathrm{d}x =...

The lessons show using Permutations and Combinations for other problems in this section, but we were having difficulty what to choose to put in those formulas. I used this approach: We have 10 numeric and 6 alphabetic = 16 total possible. Let's look at the first digit numeric, and the next 3...

this is the answer Is this right?

In a park , 200 foxes are tagged. In 100 sighting, 14 were tagged. Estimate the size of the fox population ? This is how approached . 200 tagged : Population 14 tagged : 100 (200x100/14)= Estimated Population = 1,429 I wonder if this is right !

In its flip a lid contest, a coffee chain offers prizes of 50,000 free coffees, each worth \$1.50; two new TVs, each worth \$1200; a snowmobile worth \$15 000; and sports car worth \$35 000. A total of 1 000 000 promotional coffee cups have been printed for contest. Coffee sells for \$1.50 per...

ok I still don't know where the (0.2) comes from

$$P(A|B) = P(A \cap B) / P(A)$$ $$P(A) = \text{Chance of 1 being received} = .4 * .98 + .6 * .01 = .398$$ $$P(A \cap B) = \text{Chance 1 being sent and 1 being received} = .4 * .98 = .392$$ $$P(A|B) = P(A \cap B) / P(A) = .392 / .398 = .985$$ The correct answer is 147/148 ~= .9932 What am I...

My answer is as follows: Let ##S## be the set of all outcomes of dealing four labelled 13-card hands from a standard 52-card deck. Let ##A## be event "N and E have exactly the same number of spades." Let ##A_i## be event "N and E have exactly ##i## spades each." Note that when ##i > 6##...

Summary:: There's 11 fruits, 3 of which is poisionous. A guy eats 4 of them, a girl eats 6 and a dog gets the last one. What is the conditional probability of both the girl and guy dying IF the dog made it? One fruit is enough to kill you. $$P(dog lives) = 8/11$$ $$P(allPeopleDie | dog...

I have written a finite difference program to solve 1D time-independent Schrodinger equation. It seems to work correctly for harmonic oscillator, particle in a box, etc. But I can't figure out how to calculate the probability current density. It should be constant, but what is it? The program...

The odds are $$\frac {13C5 \times 2 \times 13C3 \times 13C2} {52C13}$$ This is correct according to my book. The follow up question then becomes, what if you can pick 5 of any suit, 3 of any other suit, another 3 from the remaining 2 suits and 2 from the last untouched suit? The solution...